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| − | ==Problem==
| + | #redirect [[2022 AMC 12A Problems/Problem 19]] |
| − | Suppose that 13 cards numbered <math>1, 2, 3, \cdots, 13</math> are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the <math>13!</math> possible orderings of the cards will the <math>13</math> cards be picked up in exactly two passes?
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| − | | |
| − | <asy>
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| − | size(11cm);
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| − | draw((0,0)--(2,0)--(2,3)--(0,3)--cycle);
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| − | label("7", (1,1.5));
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| − | draw((3,0)--(5,0)--(5,3)--(3,3)--cycle);
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| − | label("11", (4,1.5));
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| − | draw((6,0)--(8,0)--(8,3)--(6,3)--cycle);
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| − | label("8", (7,1.5));
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| − | draw((9,0)--(11,0)--(11,3)--(9,3)--cycle);
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| − | label("6", (10,1.5));
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| − | draw((12,0)--(14,0)--(14,3)--(12,3)--cycle);
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| − | label("4", (13,1.5));
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| − | draw((15,0)--(17,0)--(17,3)--(15,3)--cycle);
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| − | label("5", (16,1.5));
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| − | draw((18,0)--(20,0)--(20,3)--(18,3)--cycle);
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| − | label("9", (19,1.5));
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| − | draw((21,0)--(23,0)--(23,3)--(21,3)--cycle);
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| − | label("12", (22,1.5));
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| − | draw((24,0)--(26,0)--(26,3)--(24,3)--cycle);
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| − | label("1", (25,1.5));
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| − | draw((27,0)--(29,0)--(29,3)--(27,3)--cycle);
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| − | label("13", (28,1.5));
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| − | draw((30,0)--(32,0)--(32,3)--(30,3)--cycle);
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| − | label("10", (31,1.5));
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| − | draw((33,0)--(35,0)--(35,3)--(33,3)--cycle);
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| − | label("2", (34,1.5));
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| − | draw((36,0)--(38,0)--(38,3)--(36,3)--cycle);
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| − | label("3", (37,1.5));
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| − | </asy>
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| − | | |
| − | <math>\textbf{(A) }4082\qquad\textbf{(B) }4095\qquad\textbf{(C) }4096\qquad\textbf{(D) }8178\qquad\textbf{(E) }8191</math>
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| − | | |
| − | ==Solution 1 by OmegaLearn Using Combinatorial Identities and Overcounting==
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| − | | |
| − | https://youtu.be/gW8gPEEHSfU
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| − | | |
| − | ~ pi_is_3.14
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| − | | |
| − | == See Also ==
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| − | | |
| − | {{AMC10 box|year=2022|ab=A|num-b=21|num-a=23}}
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| − | {{MAA Notice}}
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